Paper presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Malta, 16-18 July, 2012.
The development of ﬂuid motion in an inﬁnitely long circular pipe with homogeneously distributed internal heat source is examined numerically. The pipe is placed vertically in the gravity ﬁeld with the pipe wall temperature being kept constant. The motion of the ﬂuid is driven upward by the buoyancy force as well as downward by an applied pressure gradient along the pipe axis. Thus, the basic velocity proﬁle can become inﬂectional and we may anticipate that the ﬂow may become unstable in contrast to the isothermal pipe ﬂow which is known to be linearly stable for any Reynolds number. We ﬁnd that the linear instabilities always occur within the region where the basic velocity proﬁle is inﬂectional but not totally reverse. Our nonlinear analysis indicates that there are two types of nonlinear solutions, referred to as spirals and ribbons. They bifurcate simultaneously from the same point on the neutral curve. Furthermore, the branch of the ribbon extends far inside the region where the basic state is linearly stable and reaches the isothermal limit, creating a nonlinear solution in ’pure’ pipe ﬂow for the case with Pr = 0. For the case with Pr = 7 nonlinear interactions between spirals with different azimuthal wavenumbers are observed.